"The secret to breaking the parallel is pulling yourself down with your hip flexors. Here is how to learn this skill. Lie on your back with your legs straight. Place your hands on your hip flexors, right below your 'lower abs'. Have your training partner hold on to your ankles and provide some resistance. Arch your lower back - the opposite of a crunch - and press your tailbone into the deck. Slowly pull your knees all the way up to your chest against your buddy's resistance. Not the hip flexor contraction.It's an interesting observation; that good squatters actively "pull" themselves into position. There are a lot of options to try this yourself. The first time I tried it was after seeing an older PLUSA video in which Bernie Gerard (I think) recommended "reverse squats" using inversion boots. It's pretty easy to accomplish with a "Power Wheel", bands, TRX, or even just hanging weights off the feet and using a chin-up bar though grip might be a limiting factor there.
When you squat recreate the above sensation: actively pull yourself down and back with your hip flexors instead of passively yielding to the gravity. You will instantly go deeper, improve your control of the weight, and tighten up the arch in your lower back. This results in a bigger, deeper, and safer squat."
- from Beyond Bodybuilding by Pavel Tsatsouline
I've spoken with a few people who just don't "get" the hips active role in squatting, and for those who, in spite of all experimentation, can't "get it", I would suggest the following - wear a heavy ruck and go for a long march with a few hills thrown in. The next day, warm up well and then try a light-moderate squatting session - it should be all the "activation drill" needed to understand how involved the hip flexors are in a heavy squat.